Electromagnetic cloaking method

ABSTRACT

A method of constructing a concealing volume comprises constructing a plurality of concealing volume elements around a concealable volume. Each concealing volume element has a material parameter arranged to direct a propagating wave around the concealable volume. The material parameter can be refractive index, electrical permittivity, and magnetic permittivity. The concealing volume can be a metamaterial. The concealing volume diverts incoming propagating waves such that outgoing propagating waves appear to be unperturbed to an observer.

CROSS-REFERENCE TO RELATED APPLICATIONS

The present application is a continuation of U.S. patent applicationSer. No. 11/459,728, filed on Jul. 25, 2006, the entirety of which isincorporated herein by reference.

BACKGROUND

In order to exploit electromagnetism, materials are commonly used tocontrol and direct the electromagnetic fields. For example, a glass lensin a camera directs the rays of light to form an image, metal cages areused to screen sensitive equipment from electromagnetic radiation, andvarious forms of ‘black bodies’ are utilized to prevent unwantedreflections. One aspect of electromagnetism which is of particularinterest, is the use of materials in the manipulation of electromagneticwaves such as to conceal or cloak objects or volumes of space fromdetection by an outside observer.

Several known methods exist which attempt to achieve electromagneticconcealment of objects. For example, it is possible to use a series ofcameras to project an image to an observer of what he would see if anobject in question were not blocking his view path. As a result, theobserver does not realize that the object is present. This method,however, relies on the use of active components, and depends heavily onthe relative positioning of the object, cameras and observer at alltimes.

Further known concealment methods include traditional “stealthtechnology” and the use of low radar cross section structures. These aredesigned to minimize back reflection of radar or other electromagneticwaves. Whilst these structures can provide a reduced or alteredelectromagnetic signature, because they involve either the scattering ofincident waves away from the target or absorbing incident waves, theobjects which they hide are still detectable in transmission.

In their paper, Physics Rev. E Vol 72, Art. No., 016623 (2005), A Aluand N Engheta suggest a scheme for the concealment of spherical andcylindrical objects through the use of plasmonic and metamaterial‘cloaks’ or covers. Whilst this paper provides a method of reducing thetotal scattering cross section of such objects, it relies on a specificknowledge of the shape and material properties of the object beinghidden. In particular, the electromagnetic cloak and the concealedobject form a composite, whose scattering properties can be reduced inthe lowest order approximation. Therefore, if the shape of the objectchanges, the shape of the cloak must change accordingly. Furthermore,this method relies on a resonance effect, such that if the frequencydrifts away from its resonant peak, the method is less effective. It istherefore a narrowband method, which cannot be implemented for broadbandapplications.

A further aspect of electromagnetism which is of interest is the use ofmaterials in electromagnetic sensing and energy harvesting applications.Several known devices exist in this area, such as satellite dishes andsolar energy panels. Whilst such prior art devices are operable tocollect or detect electromagnetic radiation incident upon them from anumber of different directions, and can be moveable to capture radiationincident from any desired direction, they do not have the capability tocapture electromagnetic radiation incident from all directions at anygiven time. Problems therefore arise in applications when the directionof the electromagnetic source is initially unknown or constantlychanging, such as in solar energy collection and microwave energybeaming on mobile platforms.

SUMMARY

The invention is set out in the claims. According to a first embodiment,because a method is provided in which those rays which would have passedthrough a particular volume of space are deflected around the volume andreturned to their original trajectory, an observer would conclude thatthe rays had passed directly through that volume of space. This will bethe case regardless of the relative positioning of the observer and theconcealed volume. Furthermore, because no radiation can get into theconcealed volume nor any radiation gets out, an object of any shape ormaterial placed in the concealed volume will be invisible to theobserver.

In one aspect, the invention utilizes a co-ordinate transformationapproach which is independent of the sizes of the concealed andconcealing volume and does not suffer from any of the fundamentalscaling issues which affect known concealment schemes. It is alsopossible to use the transformation method for any shape of concealed orconcealing volume. The wavelength of the electromagnetic radiation doesnot appear in the solution, such that the invention can be applied forany size structure and for any wavelength, and for near and far fields.

In a further embodiment of the invention, all electromagnetic fieldswhich are incident on an outer surface of a particular volume of spaceare concentrated into a inner core region, regardless of their directionof incidence on the outer surface. This enables a detector or collectorto be placed at the inner core into which the electromagnetic fieldshave been concentrated, allowing it to interact with the intensifiedfields from all directions at that location. Because a method isdescribed in which those rays which pass through the particular volumeof space are returned to their original trajectory outwards of its outersurface, an observer would conclude that the rays had passed directlythrough that volume of space, regardless of the relative positioning ofthe observer and the volume in question. In addition, the presentinvention enables a detector placed within the inner core region to haveits material properties matched to the surrounding layer such that itwill also be invisible to an observer viewing the set up from anydirection.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is a cross-sectional ray trajectory diagram for a sphere ofradius r<R₁ cloaked within an annulus of radius R₁<r<R₂, assuming thatR₂>>λ;

FIG. 2a shows a field line in free space plotted on a Cartesian mesh;

FIG. 2b shows the field line and background coordinates of FIG. 2a bothdistorted by the same transformation;

FIG. 2c shows a small element resembling a parallelpiped;

FIG. 2d shows an integration path for solving Maxwell's equations;

FIG. 3 is a cross-sectional view of the electrostatic displacement fieldlines for a sphere of radius r<R₁ cloaked within an annulus of radiusR₁<r<R₂, located close to a point charge, assuming that R₂<<λ;

FIG. 4a is a three-dimensional depiction of the diversion ofelectromagnetic rays around a cloaked volume, according to the presentinvention;

FIG. 4b is a three-dimensional depiction of the diversion ofelectromagnetic rays around a cloaked surface, according to the presentinvention;

FIG. 5 shows a front perspective view of a ‘Swiss-roll’ metamaterial;

FIG. 6 shows a top view of the ‘Swiss-roll’ metamaterial of FIG. 5;

FIG. 7 shows a single split ring metamaterial;

FIG. 8 shows a plan view of a split ring structure in a square array,lattice spacing α;

FIG. 9 depicts the building of three-dimensional split ring resonator(SRR) at unit cell level;

FIG. 10 is a prior-art negative index metamaterial comprising SRR's andwires;

FIG. 11 is a cross-sectional ray trajectory diagram of a concentrationdevice of radius R₃ in which a spherical volume of space radius R₂ iscompressed to a smaller volume of space radius R₁; and

FIG. 12 is a cross-sectional ray trajectory diagram in which rays from apoint source incident on a sphere radius R₃ are concentrated into aninner sphere radius R₁.

FIG. 13 is a flow chart depicting the steps of method 1300: identifyinga coordinate transform (1302); computing the material parameters (1304);selecting diverting volume elements (1306); and constructing thediverting volume (1308).

DETAILED DESCRIPTION

In overview, according to a first embodiment the invention provides amethod of concealing an object or volume of space by means ofredirecting the electromagnetic field lines around it. Any field lineswhich normally would flow through the concealed volume are redirectedthrough a surrounding concealing volume. Outside of the cloakingstructure, the field lines remain unperturbed. The cloaking structure isformed from a metamaterial, that is a material designed to exhibit thespecific material parameters required for cloaking the object or volumein question. In particular, the electric permittivity ∈ and magneticpermeability μ and hence the refractive index n of the cloakingmetamaterial can be varied effectively continuously spatially in orderto manipulate the electromagnetic field lines and hence effectivelyconceal the enclosed volume. As a result of the invention, the concealedvolume is undetectable to an outside observer, and any electromagneticradiation produced by an object within the concealed volume cannotescape from it.

In a further embodiment the invention provides a method of concentratingthe electromagnetic fields incident upon the outer surface of aconcentrator device into an inner core region. The concentratorstructure comprises a concentrating volume and a collecting volume. Anyfield lines which would normally flow through the concentrating volumeare redirected into an inner collecting volume. Outside of theconcentrator annulus surrounding the concentrating volume, the fieldlines remain unperturbed. The concentrator structure is formed from ametamaterial in a similar manner to that described above for thecloaking structure above.

Whilst, as is outlined in more detail below, the transformation methoddescribed herein can be applied to conceal any desired volume using anydesired cloaking structure, for simplicity it will first be described inrelation to the simple spherical case as shown in FIG. 1. According toFIG. 1, the volume to be hidden is a sphere 100 of radius R₁ and thecloaking material is contained within the annulus 102 of radius R₁<r<R₂.In order to create a secure, radiation-free volume within the sphere 100of radius R₁, all those electromagnetic field lines 104 which normallywould flow through the sphere 106 of radius r<R₂ must therefore becompressed into the cloaking annulus 102 R₁<r<R₂. That is, in order toremove the sphere 100 of radius R₁ from view, the volume within thatsphere 100 plus the volume within the annulus 102 must all be mappedonto the annulus 102 of radius R₁<r<R₂. In the medium outside of thesphere 106 r<R₂, the electromagnetic field lines 104 must remainunperturbed.

The theory of propagation of electromagnetic waves is well known and soonly the relevant aspects are summarised here in the context of theinvention. In free space, the magnetic field strength H is related tothe magnetic field intensity B and the magnetic permeability of freespace, μ₀, via the equation B=μ₀H. The electric field E in free space isrelated to the electric displacement field D and electric permittivityof free space ∈₀ via the equation D=∈₀E. In a dielectric the value of ∈and μ are modified by the relative permittivity and permeability value∈_(r), μ_(r) and the refractive index of the invention is given byn_(r)=^(±)√{square root over (∈μ)}. As will be well known to the skilledperson in the art, the three vectors E, H, and S are mutuallyorthogonal, with the Poynting vector S being calculable via the vectorcross product S=E×H. The Poynting vector S gives the direction andmagnitude of energy flow per unit area per unit time of anelectromagnetic wave, and in a ray trajectory diagram such as that shownin FIG. 1, the direction of the rays of light correspond to theelectromagnetic field lines and follow the direction of the Poyntingvector. In order to implement this invention therefore, it is necessaryto focus or redirect any of the electric displacement field D, themagnetic field intensity B, and the Poynting vector S so that they avoidthe sphere 100 of radius R₁, and return undisturbed to their originaltrajectories.

As described in more detail below the approach further requires solutionof Maxwell's equations in relation to the redirected fields. For ease ofdepiction, redirection of the Poynting vector in particular will beaddressed in the following description.

In order to calculate the electromagnetic properties which a cloakingstructure must possess in order to effectively implement the desiredcloaking effects, the first step is to calculate the distortion whichmust be applied to the Poynting vector S so that the rays do not passthrough the sphere 100 of radius R₁. As shown in FIGS. 2a and 2b , thisis achieved by recording the initial configuration of each ray, whichwould occur in the absence of a cloaked volume and cloaking material, ona known geometric mesh, such as a spherical or Cartesian mesh. Forexample in FIG. 2a , a single ray 200 is plotted on a Cartesian mesh202: Once the appropriate rays 200 have been plotted on it, theCartesian mesh 202 is stretched or distorted in such a manner that therays plotted thereon become distorted to follow their new desired path,avoiding the cloaked volume. The distortions which are imposed on theCartesian mesh 202 in order to achieve this can then be recorded as acoordinate transformation between the original Cartesian mesh 202 and asecond transformed mesh 204, wherein the second mesh is defined by:

q ₁(x,y,z),q ₂(x,y,z),q ₃(x,y,z)  (1)

In this second mesh 204, lines of constant q₂, q₃ define the generalizedq₁ axis, and so on. It will be seen that conversely if the second mesh204 comprised a set of points defined by equal increments along the q₁,q₂, q₃ axes, it would appear distorted in the x, y, z Cartesian frame,as a comparison of FIGS. 2a and 2b clearly demonstrates.

The benefit of performing this transformation is that, as describedbelow, it can be shown that it is possible to solve Maxwell's equationsin the same form in the transformed geometry as in the originalgeometry, with only a corresponding transform of the spatiallydistributed values of ∈ and μ being required. This can be understoodintuitively from the recognition that variation of the permittivity Eand permeability μ values, and hence the refractive index n in a space,is equivalent to distorting the geometry of the space as seen by a lightray. It will be noted that the approach specifically addressesmodification of the value of ∈ and μ, individually rather than of therefractive index n=+±√{square root over (∈μ)}. In particular this allowsreflection between interfaces of different media to be removed byensuring impedance (Z) matching where

$Z = {\sqrt{\frac{\mu}{ɛ}}.}$

As will be known to the skilled person, Maxwell's equations in a systemof Cartesian coordinates take the form:

∇×E=−μμ ₀ ∂H/∂t,

∇×E=−∈∈ ₀ ∂E/∂t  (2)

where both electric permittivity ∈ and magnetic permeability μ depend onposition. When a coordinate transformation is applied to theseequations, in the new co-ordinate system they take the form:

∇_(q) ×Ê=−μ ₀ {circumflex over (μ)}∂Ĥ/∂t,

∇_(q) ×Ĥ=−∈ ₀ {circumflex over (∈)}∂Ê/∂t  (3)

where ∈ and μ are in general tensors, and E, and H are renormalizedelectric and magnetic fields. All four quantities are simply related tooriginals. In other words the form of Maxwell's equation is preserved bya co-ordinate transformation: the co-ordinate transformation does notchange the fact that we are still solving Maxwell's equations, it simplychanges the definition of ∈, μ.

In particular, the effect of the transformation of μ and ∈ is to scalethem both by a common factor. A generalized solution to this is asfollows.

We define three units vectors, u₁, u₂, u₃, to point along thegeneralized q₁, q₂, q₃ axes. The length of a line element is given by,

ds ² =dx ² +dy ² +dz ² =Q ₁₁ dq ₁ ² +Q ₂₂ M+Q ₃₃ dq ₃ ²+2Q ₁₂ dq ₁ dq₂+2Q ₁₃ dq ₁ dq ₃+2Q ₂₃ dq ₂ dq ₃  (G1)

where,

$\begin{matrix}{Q_{ij} = {{\frac{\partial x}{\partial q_{i}}\frac{\partial x}{\partial q_{j}}} + {\frac{\partial y}{\partial q_{i}}\frac{\partial y}{\partial q_{j}}} + {\frac{\partial z}{\partial q_{i}}\frac{\partial z}{\partial q_{j}}}}} & ({G2})\end{matrix}$

In particular we shall need the length of a line element directed alongone of the three axes,

ds _(i) =Q _(i) dq _(i)  (G3)

where for shorthand,

Q _(i) ² =Q _(ii)  (G4)

To calculate ∇×E consider a small element, small enough that itresembles a parallelepiped (FIG. 2c ). In this we assume that thetransformation has no singularities such as points or lines where theco-ordinate system suddenly heads off in a different direction.First we calculate the projection of ∇×E onto the normal to the u₁-u₂plane by taking a line integral around the u₁-u₂ parallelogram andapplying Stokes' theorem (FIG. 3). We define,

E ₁ =E·u ₁ ,E ₂ =E·u ₂ ,E ₃ =E·u ₃  (G5)

so that,

$\begin{matrix}{{{( {\nabla{\times E}} ) \cdot ( {u_{1} \times u_{2}} )}d\; q_{1}Q_{1}d\; q_{2}Q_{2}} = {{d\; q_{1}\frac{\partial}{\partial q_{1}}( {E_{2}{dq}_{2}Q_{2}} )} - {{dq}_{2}\frac{\partial}{\partial q_{2}}( {E_{1}d\; q_{1}Q_{1}} )}}} & ({G6})\end{matrix}$

or,

$\begin{matrix}{{{( {\nabla{\times E}} ) \cdot ( {u_{1} \times u_{2}} )}Q_{1}Q_{2}} = {{\frac{\partial{\hat{E}}_{2}}{\partial q_{1}} - \frac{\partial{\hat{E}}_{1}}{\partial q_{2}}} = ( {\nabla_{q}{\times \hat{E}}} )^{3}}} & ({G7})\end{matrix}$

where we use the conventional superscript notation for contravariantcomponents of a vector. We have defined,

Ê ₁ =Q ₁ E ₁ ,Ê ₂ =Q ₂ E ₂ ,Ê ₃ =Q ₃ E ₃  (G8)

Note that the right-hand side of equation (10) is simple ‘component 3’of curl evaluated in the new co-ordinate system. Now applying Maxwell,

$\begin{matrix}{{{( {\nabla{\times E}} ) \cdot ( {u_{1} \times u_{2}} )}Q_{1}Q_{2}} = {{- \mu_{0}}\mu {\frac{\partial H}{\partial t} \cdot ( {u_{1} \times u_{2}} )}Q_{1}Q_{2}}} & ({G9})\end{matrix}$

We write H in terms of the contravariant components,

H=H ¹ u ₁ +H ² u ₂ +H ³ u ₃  (G10)

which in turn can be expressed in terms of the covariant components,

$\begin{matrix}{{g^{- 1}\begin{bmatrix}H^{1} \\H^{2} \\H^{3}\end{bmatrix}} = {{\begin{bmatrix}{u_{1} \cdot u_{1}} & {u_{1} \cdot u_{2}} & {u_{1} \cdot u_{3}} \\{u_{2} \cdot u_{1}} & {u_{2} \cdot u_{2}} & {u_{2} \cdot u_{3}} \\{u_{3} \cdot u_{1}} & {u_{3} \cdot u_{2}} & {u_{3} \cdot u_{3}}\end{bmatrix}\begin{bmatrix}H^{1} \\H^{2} \\H^{3}\end{bmatrix}} = \begin{bmatrix}H_{1} \\H_{2} \\H_{3}\end{bmatrix}}} & ({G11})\end{matrix}$

where the first part defines g, and

H ₁ =H·u ₁ ,H ₂ =H·u ₂ ,H ₃ =H·u ₃  (G12)

Inverting g gives

H ^(i)=Σ_(j=1) ³ g ^(ij) H _(j).  (G13)

Substituting equation (13) into (12) gives

$\begin{matrix}{{{( {\nabla{\times E}} ) \cdot ( {u_{1} \times u_{2}} )}Q_{1}Q_{2}} = {{{- \mu_{0}}\mu {\frac{\partial H}{\partial t} \cdot ( {u_{1} \times u_{2}} )}Q_{1}Q_{2}} = {{- \mu_{0}}\mu {\sum\limits_{j = 1}^{3}{g^{3j}\frac{\partial H_{j}}{\partial t}{u_{3} \cdot ( {u_{1} \times u_{2}} )}Q_{1}Q_{2}}}}}} & ({G14})\end{matrix}$

Define

{circumflex over (μ)}^(ij) =μg ^(ij) |u ₁·(u ₂ ×u ₃)|Q ₁ Q ₂ Q ₃(Q _(i)Q _(j))⁻¹  (G15)

And

Ĥ _(j) =Q _(j) H _(j)  (G16)

so that,

$\begin{matrix}{{{( {\nabla{\times E}} ) \cdot ( {u_{1} \times u_{2}} )}Q_{1}Q_{2}} = {{- \mu_{0}}{\sum\limits_{j = 1}^{3}{{\hat{\mu}}^{3j}\frac{\partial{\hat{H}}_{j}}{\partial t}}}}} & ({G17})\end{matrix}$

Hence on substituting from equation (10),

$\begin{matrix}{( {\nabla_{q}{\times \hat{E}}} )^{i} = {{- \mu_{0}}{\sum\limits_{j = 1}^{3}{{\hat{\mu}}^{ij}\frac{\partial{\hat{H}}_{j}}{\partial t}}}}} & ({G18})\end{matrix}$

and by symmetry between E and H fields,

$\begin{matrix}{( {\nabla_{q}{\times \hat{H}}} )^{i} = {{+ ɛ_{0}}{\sum\limits_{j = 1}^{3}{{\hat{ɛ}}^{ij}\frac{\partial{\hat{E}}_{j}}{\partial t}}}}} & ({G19})\end{matrix}$

where

{circumflex over (∈)}^(ij) =∈g ^(ij) |u ₁·(u ₂ ×u ₃)|Q ₁ Q ₂ Q ₃(Q _(i)Q _(j))⁻¹  (G20)

Note that these expressions simplify considerable if the new co-ordinatesystem is orthogonal, e.g., cylindrical or spherical, when

g ^(ij) |u ₁·(u ₂ ×u ₃)|=δ_(ij)  (G21)

In the case of an orthogonal geometry (Cartesian, spherical orcylindrical) but the renormalized values of the permittivity andpermeability are:

$\begin{matrix}{{ɛ_{u}^{\prime} = {ɛ_{u}\frac{Q_{u}Q_{v}Q_{w}}{Q_{u}^{2}}}},{\mu_{u}^{\prime} = {\mu_{u}\frac{Q_{u}Q_{v}Q_{w}}{Q_{u}^{2}}}},{{{etc}.E_{u}^{\prime}} = {Q_{u}E_{u}}},{H_{u}^{\prime}\; = {Q_{u}H_{u}}},{{etc}.}} & (4) \\{{where},} & (5) \\{{Q_{u}^{2} = {( \frac{\partial x}{\partial u} )^{2} + ( \frac{\partial y}{\partial u} )^{2} + ( \frac{\partial z}{\partial u} )^{2}}}{Q_{v}^{2} = {( \frac{\partial x}{\partial v} )^{2} + ( \frac{\partial y}{\partial v} )^{2} + ( \frac{\partial z}{\partial v} )^{2}}}{Q_{w}^{2} = {( \frac{\partial x}{\partial w} )^{2} + ( \frac{\partial y}{\partial w} )^{2} + ( \frac{\partial z}{\partial w} )^{2}}}} & (6)\end{matrix}$And:

B′=μ ₀ μ′H′ and D′=∈ ₀ ∈E′  (7)

Referring back to the spherical case as shown in FIG. 1, the naturalchoice of mesh on which to plot the original ray configuration is aspherical one, which uses the co-ordinates r, θ and φ. The co-ordinatetransformation operates by taking all the fields in the sphere 106 ofradius r<R₂ and compressing them into the annulus 102 of radius R₁<r<R₂.This is achieved by any transformation of the form:

$\begin{matrix}{{r^{\prime} = {R_{1} + {\frac{{f(r)} - {f(0)}}{{f( R_{2} )} - {f(0)}}( {R_{2} - R_{1}} )}}},{\theta^{\prime} = \theta},{\varphi^{\prime} = \varphi}} & (8)\end{matrix}$

where f(r) is a monotonically increasing function of r, this will mapthe entire interior of a sphere 106 of radius R₂ into an annulus boundedby two spheres 100, 106 radii R₁ and R₂ respectively. There is symmetryfor φ and θ because the field lines are being compressed radially only,so their angular positioning is not distorted.

The simplest solution to equation (8) arises when f(r)=r, whichsimplifies to give:

r′=R ₁ +r(R ₂ −R ₁)/R ₂,

θ′=θ,

φ′=φ  (9)

In order to ascertain the electromagnetic properties which will berequired of the cloaking annulus 102 material so that it achieves thedesired result of cloaking any object in the sphere 100 R₁, thetransformation which has been imposed on r, φ and θ must be applied inan analogous manner to ∈ and μ, as shown for the general case above. Ateach point (x, y, z) in space, ∈ and μ will have component (∈_(x),∈_(y), ∈_(z)), (μ_(x), μ_(y), μ_(z)) which must be transformed tospherical coordinate valve (∈′_(r), ∈′_(θ), ∈′_(φ)), (μ′_(r), μ′_(θ),μ′_(φ)) in the transformed geometry.

We do this in two steps:

First transform to spherical coordinates without any compression,

x=r cos φ sin θ,y=r sin φ sin θ,Z=r cos θ  (A1)

so that, as shown in equation (4) above, in the new frame,

$\begin{matrix}{{{\overset{\sim}{ɛ}}_{i} = {ɛ_{i}\frac{Q_{1}Q_{2}Q_{3}}{Q_{i}^{2}}}},{{\overset{\sim}{\mu}}_{i} = {\mu_{i}\frac{Q_{1}Q_{2}Q_{3}}{Q_{i}^{2}}}}} & ( {A\; 2} )\end{matrix}$

We calculate,

$\begin{matrix}{{Q_{r}^{2} = {{( {\frac{\partial x}{\partial r} = {\cos \; \varphi \; \sin \; \theta}} )^{2} + ( {\frac{\partial y}{\partial r} = {\sin \; \varphi \; \sin \; \theta}} )^{2} + ( {\frac{\partial z}{\partial r} = {\cos \; \theta}} )^{2}} = 1}}{Q_{\theta}^{2} = {{( {\frac{\partial x}{\partial\theta} = {r\; \cos \; \varphi \; \cos \; \theta}} )^{2} + ( {\frac{\partial y}{\partial\theta} = {r\; \sin \; \varphi \; \cos \; \theta}} )^{2} + ( {\frac{\partial z}{\partial\theta} = {{- r}\; \sin \; \theta}} )^{2}} = r^{2}}}{Q_{\varphi}^{2} = {{( {\frac{\partial x}{\partial\varphi} = {{- r}\; \sin \; \varphi \; \sin \; \theta}} )^{2} + ( {\frac{\partial y}{\partial Z} = {{+ r}\; \cos \; \varphi \; \sin \; \theta}} )^{2} + ( {\frac{\partial z}{\partial Z} = 0} )^{2}} = {r^{2}\sin^{2}\theta}}}} & ({A3})\end{matrix}$

and hence,

$\begin{matrix}{{{\overset{\sim}{ɛ}}_{r} = {{\overset{\sim}{\mu}}_{r} = {{\frac{Q_{r}Q_{\theta}Q_{\varphi}}{Q_{r}^{2}}ɛ_{r}} = {{\frac{r^{2}\sin \; \theta}{1}ɛ_{r}}\; = {r^{2}\sin \; \theta \; ɛ_{r}}}}}}{{\overset{\sim}{ɛ}}_{\theta} = {{\overset{\sim}{\mu}}_{\theta} = {{\frac{Q_{r}Q_{\theta}Q_{\varphi}}{Q_{\theta}^{2}}ɛ_{\theta}} = {{\frac{r^{2}\sin \; \theta}{r^{2}}ɛ_{\theta}}\; = {\sin \; \theta \; ɛ_{\theta}}}}}}{{\overset{\sim}{ɛ}}_{\varphi} = {{\overset{\sim}{\mu}}_{\varphi} = {{\frac{Q_{r}Q_{\theta}Q_{\varphi}}{Q_{\varphi}^{2}}ɛ_{\varphi}} = {{\frac{r^{2}\sin \; \theta}{r^{2}\sin^{2}\theta}ɛ_{\varphi}}\; = {\frac{1}{\sin \; \theta}ɛ_{\varphi}}}}}}} & ({A4})\end{matrix}$

where ∈_(r), ∈_(θ), ∈_(φ) the three components of the permittivitytensor in the original Cartesian frame, and we assume that thepermittivity and permeability are equal. Note that we can easily extractthe Cartesian view of ∈_(r), ∈_(θ), ∈_(φ) from (A4) by removing theappropriate factors of r.

In order to create a protected space inside the sphere radius R₁ we makea further transformation to a new cylindrical system in which the radiusis compressed taking any ray trajectories with it. It is noted thatcomputer programs work in Cartesian coordinates and therefore that it ispossible to re-express the compressed radial coordinate system in termsof a compressed Cartesian system, x′y′z′, by removing factors of r′.

Consider the transformation from equation (9)

r′=R ₁ +r(R ₂ −R ₁)/R ₂,θ′=θ,φ′=φ  (A5)

which can be re-expressed as,

$\begin{matrix}{{r = \frac{( {r^{\prime} - R_{1}} )R_{2}}{R_{2} - R_{1}}},{\theta^{\prime} = \theta},{\varphi^{\prime} = \varphi}} & ({A6})\end{matrix}$

We calculate,

$\begin{matrix}{{Q_{r^{\prime}}^{2} = {{( {\frac{\partial r}{\partial r^{\prime}} = \frac{R_{2}}{R_{2} - R_{1}}} )^{2} + ( {\frac{\partial\theta}{\partial r^{\prime}} = 0} )^{2} + ( {\frac{\partial\varphi}{\partial r^{\prime}} = 0} )^{2}} = \lbrack \frac{R_{2}}{R_{2} - R_{1}} \rbrack^{2}}}\mspace{20mu} {Q_{\theta^{\prime}}^{2} = {{( {\frac{\partial r}{\partial\theta^{\prime}} = 0} )^{2} + ( {\frac{\partial\theta}{\partial\theta^{\prime}} = 1} )^{2} + ( {\frac{\partial\varphi}{\partial\theta^{\prime}} = 0} )^{2}} = 1}}\mspace{20mu} {Q_{\varphi^{\prime}}^{2} = {{( {\frac{\partial r}{\partial\varphi^{\prime}} = 0} )^{2} + ( {\frac{\partial\theta}{\partial\varphi^{\prime}} = 0} )^{2} + ( {\frac{\partial\varphi}{\partial\varphi^{\prime}} = 1} )^{2}} = 1}}} & ({A7})\end{matrix}$

so that in the new frame,for R₁<r′<R₂

$\begin{matrix}{\begin{matrix}{{\overset{\sim}{ɛ}}_{r^{\prime}} = {\overset{\sim}{\mu}}_{r^{\prime}}} \\{= {{\overset{\sim}{ɛ}}_{r}\frac{Q_{r^{\prime}}Q_{\theta^{\prime}}Q_{\varphi^{\prime}}}{Q_{r^{\prime}}^{2}}}} \\{= {\frac{( {R_{2} - R_{1}} )}{R_{2}}r^{2}\sin \; {\theta ɛ}_{r}}} \\{= {{\frac{( {R_{2} - R_{1}} )}{R_{2}}\lbrack \frac{( {r^{\prime} - R_{1}} )R_{2}}{R_{2} - R_{1}} \rbrack}^{2}\sin \; {\theta ɛ}_{r}}} \\{= {\frac{{R_{2}( {r^{\prime} - R_{1}} )}^{2}}{R_{2} - R_{1}}\sin \; {\theta ɛ}_{r}}}\end{matrix}{{\overset{\sim}{ɛ}}_{\theta^{\prime}} = {{\overset{\sim}{\mu}}_{\theta^{\prime}} = {{{\overset{\sim}{ɛ}}_{\theta}\frac{Q_{r^{\prime}}Q_{\theta^{\prime}}Q_{\varphi^{\prime}}}{Q_{\theta^{\prime}}^{2}}} = {\frac{R_{2}}{R_{2} - R_{1}}\sin \; \theta \; ɛ_{\theta}}}}}{{\overset{\sim}{ɛ}}_{\varphi^{\prime}} = {{\overset{\sim}{\mu}}_{\varphi^{\prime}} = {{{\overset{\sim}{ɛ}}_{\varphi}\frac{Q_{r^{\prime}}Q_{\theta^{\prime}}Q_{\varphi^{\prime}}}{Q_{\varphi^{\prime}}^{2}}} = {\frac{R_{2}}{R_{2} - R_{1}}\frac{1}{\sin \; \theta}ɛ_{\varphi}}}}}} & ({A8a})\end{matrix}$

Alternatively we can reinterpret these values in a Cartesian coordinateframe, x′ y′ z′, which is easily done by comparison with (A4),

$\begin{matrix}{{ɛ_{r^{\prime}} = {\mu_{r^{\prime}} = {\frac{R_{2}}{R_{2} - R_{1}}\frac{( {r^{\prime} - R_{1}} )^{2}}{r^{\prime 2}}ɛ_{r}}}}{ɛ_{\theta^{\prime}} = {\mu_{\theta^{\prime}} = {\frac{R_{2}}{R_{2} - R_{1}}ɛ_{\theta}}}}{{\overset{\sim}{ɛ}}_{\varphi^{\prime}} = {{\overset{\sim}{\mu}}_{\varphi^{\prime}} = {\frac{R_{2}}{R_{2} - R_{1}}ɛ_{\varphi}}}}} & (10)\end{matrix}$

where we have assumed that the starting material is vacuum.

for R₂<r′

$\begin{matrix}{{{\overset{\sim}{ɛ}}_{r^{\prime}} = {{\overset{\sim}{\mu}}_{r^{\prime}} = {r^{\prime \; 2}\sin \; \theta^{\prime}ɛ_{r^{\prime}}}}}{{\overset{\sim}{ɛ}}_{\theta^{\prime}} = {{\overset{\sim}{\mu}}_{\theta^{\prime}} = {\sin \; \theta^{\prime}ɛ_{\theta^{\prime}}}}}{{\overset{\sim}{ɛ}}_{Z^{\prime}} = {{\overset{\sim}{\mu}}_{Z^{\prime}} = {\frac{1}{\sin \; \theta^{\prime}}ɛ_{\varphi^{\prime}}}}}} & ({A7b})\end{matrix}$In free space,∈_(r)′=μ_(r)′=∈_(θ)′=μ_(θ)′=∈_(φ)′=μ_(φ)′=1  (11)

for r′<R₁ we may choose any value we please for the dielectric functionssince radiation never penetrates into this region. In this region ∈′, μ′are free to take any value without restriction whilst still remaininginvisible, and not contributing to electromagnetic scattering. This willbe the case for all cloaked volumes, regardless of their shape or size.

If a suitable cloaking material can be implemented for the annulus 102of radius R₁<r<R₂, this will exclude all fields from entering thecentral region 100 of radius R₁ and, conversely, will prevent all fieldsfrom escaping from this region. The cloaking structure itself and anyobject placed in the concealed sphere 100 R₁ will therefore beundetectable in either reflection or transmission. It will be noted thatat all points in the medium the impedance

$z = {\sqrt{\frac{\mu}{ɛ}} = 1}$

such that no unwanted reflection in the medium, caused by impedancemismatch, takes place.

It is noted that if the impedance z=√{square root over (μ/∈)}=1 isachieved at all points in the metamaterial, the cloaking structure andconcealed volume will appear to contain only free space, and so will beinvisible if embedded in free space. If the cloaking structure isembedded in another medium the effect will differ, for example ifembedded in water it would have the appearance of a bubble. In order tomake the cloaking structure invisible in another medium it is necessaryto match the ratio of μ and ∈ in the metamaterial to that of thesurrounding medium at the interface between the two. As the skilledperson will appreciate, whilst this changes the numerical values of ∈and μ in the cloaking structure metamaterial, the general transformationtheory outlined above remains the same. That is, because the presentinvention enables ∈ and μ to be controlled in a metamaterialindependently of one another, it is possible to achieve any desiredvalue of z, to match the impedance of the metamaterial to that of itssurroundings.

Whilst in FIG. 1 the cloaking annulus 102 is depicted as beingrelatively thick, this does not have to be the case. The solution is notdependent on shell thickness or any other length scales, such that itdoes not suffer from any scaling issues. Additionally, the solutionprovided holds in both in the near and far fields, because the solutiondoes not depend on the wavelength λ. This is demonstrated in FIGS. 1 and3. For purposes of illustration, suppose that R₂>>λ where λ is thewavelength so that we can use the ray approximation to plot the Poyntingvector. If our system is then exposed to a source of radiation atinfinity we can perform the ray tracing exercise shown in FIG. 1. Rays104 in this figure result from numerical integration of a set ofHamilton's equations obtained by taking the geometric limit of Maxwell'sequations with anisotropic, inhomogeneous media. This integrationprovides an independent confirmation that the configuration specified by(9) and (10) excludes rays from the interior region 100. Alternativelyif R₂>>λ and we locate a point charge 306 nearby, the electrostatic (ormagnetostatic) approximation applies. A plot of the local electrostaticdisplacement field which results is shown in FIG. 3.

The theoretical solution outlined above can be demonstrated using aray-tracing program to calculate the ray trajectories. According to sucha program, a specified material is assumed to exist around a centralvolume that is to be ‘cloaked’, or rendered invisible to the user. Usingthe specified parameters as input, the ray-tracing code confirms thetheoretical prediction. Ray tracing programs of this type will be wellknown to those skilled in the art, although some existing programs willnot be able to deal with the complexity of the problem for thisinvention. In FIGS. 4a and 4b , a ray tracing program has been usedwhich was developed from a custom ray tracing code based on aHamiltonian formulation to solve Maxwell's equations in the geometriclimit (i.e. zero wavelength limit). The results are shown for aspherical structure 400 and a cylindrical structure 402 respectively,wherein the transformed parameters E and μ in the cylindrical case arecalculated in a similar manner to those for the sphere, as shown below:

First transform to cylindrical coordinates without any compression,

x=r cos θ,y=r sin θ,Z=z  (A8)

so that in the new frame,

$\begin{matrix}{{{\overset{\sim}{ɛ}}_{i} = {ɛ_{i}\frac{Q_{1}Q_{2}Q_{3}}{Q_{i}^{2}}}},{{\overset{\sim}{\mu}}_{i} = {\mu_{i}\frac{Q_{1}Q_{2}Q_{3}}{Q_{i}^{2}}}}} & ({A9})\end{matrix}$

We calculate,

$\begin{matrix}{{Q_{r}^{2} = {{( {\frac{\partial x}{\partial r} = {\cos \; \theta}} )^{2} + ( {\frac{\partial y}{\partial r} = {\sin \; \theta}} )^{2} + ( {\frac{\partial z}{\partial r} = 0} )^{2}} = 1}}{Q_{\theta}^{2} = {{( {\frac{\partial x}{\partial\theta} = {{- r}\; \sin \; \theta}} )^{2} + ( {\frac{\partial y}{\partial\theta} = {{+ r}\; \cos \; \theta}} )^{2} + ( {\frac{\partial z}{\partial r} = 0} )^{2}} = r^{2}}}{Q_{Z}^{2} = {{( {\frac{\partial x}{\partial Z} = 0} )^{2} + ( {\frac{\partial y}{\partial Z} = 0} )^{2} + ( {\frac{\partial z}{\partial Z} = 1} )^{2}} = 1}}} & ({A10})\end{matrix}$

and hence,

$\begin{matrix}{{{{\overset{\sim}{ɛ}}_{r} = {{\overset{\sim}{\mu}}_{r} = {{\frac{Q_{r}Q_{\theta}Q_{Z}}{Q_{r}^{2}}ɛ_{r}} = {r\; ɛ_{r}}}}}\; {{\overset{\sim}{ɛ}}_{\theta} = {{\overset{\sim}{\mu}}_{\theta} = {{\frac{Q_{r}Q_{\theta}Q_{Z}}{Q_{\theta}^{2}}ɛ_{\theta}} = {r^{- 1}ɛ_{\theta}}}}}\; {{\overset{\sim}{ɛ}}_{Z} = {{\overset{\sim}{\mu}}_{Z} = {{\frac{Q_{r}Q_{\theta}Q_{Z}}{Q_{Z}^{2}}ɛ_{Z}} = {r\; ɛ_{Z}}}}}}\;} & ({A11})\end{matrix}$

where ∈_(r), ∈_(θ), ∈_(Z) the three components of the permittivitytensor in the original Cartesian frame, and we assume that thepermittivity and permeability are equal. Note that we can easily extractthe Cartesian view of ∈_(r), ∈_(θ), ∈_(Z) from (A13) by removing theappropriate factors of r.

In order to create a protected space inside a cylinder radius R₁ we makea further transformation to a new cylindrical system in which the radiusis compressed taking any ray trajectories with it. It is noted thatcomputer programs work in Cartesian coordinates and therefore that it ispossible to re-express the compressed radial coordinate system in termsof a compressed Cartesian system, x′y′ z′, by removing factors of r′.

Consider the transformation,

r′=R ₁ +r(R ₂ −R ₁)/R ₂ ,θ′=θ,Z′=Z  (A12)

or,

$\begin{matrix}{{r = \frac{( {r^{\prime} - R_{1}} )R_{2}}{R_{2} - R_{1}}},\mspace{31mu} {\theta^{\prime} = \theta},\mspace{31mu} {Z^{\prime} = Z}} & ({A13})\end{matrix}$

We calculate,

$\begin{matrix}{{Q_{r^{\prime}}^{2} = {{( {\frac{\partial r}{\partial r^{\prime}} = \frac{R_{2}}{R_{2} - R_{1}}} )^{2} + ( {\frac{\partial\theta}{\partial r^{\prime}} = 0} )^{2} + ( {\frac{\partial Z}{\partial r^{\prime}} = 0} )^{2}} = \lbrack \frac{R_{2}}{R_{2} - R_{1}} \rbrack^{2}}}\mspace{20mu} {Q_{\theta^{\prime}}^{2} = {{( {\frac{\partial r}{\partial\theta^{\prime}} = 0} )^{2} + ( {\frac{\partial\theta}{\partial\theta^{\prime}} = 1} )^{2} + ( {\frac{\partial Z}{\partial\theta^{\prime}} = 0} )^{2}} = 1}}\mspace{20mu} {Q_{Z^{\prime}}^{2} = {{( {\frac{\partial r}{\partial Z^{\prime}} = 0} )^{2} + ( {\frac{\partial\theta}{\partial Z^{\prime}} = 0} )^{2} + ( {\frac{\partial Z}{\partial Z^{\prime}} = 1} )^{2}} = 1}}} & ({A14})\end{matrix}$

so that in the new frame,for R₁<r′<R₂

$\begin{matrix}{{{\overset{\sim}{ɛ}}_{r^{\prime}} = {{\overset{\sim}{\mu}}_{r^{\prime}} = {{{\overset{\sim}{ɛ}}_{r}\frac{Q_{r^{\prime}}Q_{\theta^{\prime}}Q_{Z^{\prime}}}{Q_{r^{\prime}}^{2}}} = {{\frac{( {R_{2} - R_{1}} )}{R_{2}}r\; ɛ_{r}} = {{\frac{( {R_{2} - R_{1}} )}{R_{2}}\frac{( {r^{\prime} - R_{1}} )R_{2}}{R_{2} - R_{1}}ɛ_{r}} = {( {r^{\prime} - R_{1}} )ɛ_{r}}}}}}}{{\overset{\sim}{ɛ}}_{\theta^{\prime}} = {{\overset{\sim}{\mu}}_{\theta^{\prime}} = {{{\overset{\sim}{ɛ}}_{\theta}\frac{Q_{r^{\prime}}Q_{\theta^{\prime}}Q_{Z^{\prime}}}{Q_{\theta^{\prime}}^{2}}} = {{\frac{R_{2}}{R_{2} - R_{1}}r^{- 1}ɛ_{\theta}} = {{\frac{R_{2}}{R_{2} - R_{1}}\frac{R_{2} - R_{1}}{( {r^{\prime} - R_{1}} )R_{2}}ɛ_{\theta}} = {\frac{1}{( {r^{\prime} - R_{1}} )}ɛ_{\theta}}}}}}}} & ({A15a}) \\{{\overset{\sim}{ɛ}}_{Z^{\prime}} = {{\overset{\sim}{\mu}}_{Z^{\prime}} = {{{\overset{\sim}{ɛ}}_{Z}\frac{Q_{r^{\prime}}Q_{\theta^{\prime}}Q_{Z^{\prime}}}{Q_{Z^{\prime}}^{2}}} = {{\frac{R_{2}}{R_{2} - R_{1}}r\; ɛ_{Z}} = {{\frac{R_{2}}{R_{2} - R_{1}}\frac{( {r^{\prime} - R_{1}} )R_{2}}{R_{2} - R_{1}}ɛ_{Z}} = {\lbrack \frac{R_{2}}{R_{2} - R_{1}} \rbrack^{2}( {r^{\prime} - R_{1}} )ɛ_{Z}}}}}}} & ({A15a})\end{matrix}$

for R₂<r′

{tilde over (∈)}_(r′)={tilde over (μ)}_(r′) =r′{tilde over (∈)} _(r′)

{tilde over (∈)}_(θ′)={tilde over (μ)}_(θ′) =r′ ⁻¹∈_(θ′)

{tilde over (∈)}_(Z′)={tilde over (μ)}_(Z′) =r′∈ _(Z′)  (A15b)

for r′<R₁ we may choose any value we please for the dielectric functionssince radiation never penetrates into this region.Alternatively we can reinterpret these values in a Cartesian coordinateframe, x′y′z′, which is easily done by comparison with (A4),

$\begin{matrix}{{ɛ_{r^{\prime}} = {\mu_{r^{\prime}} = \frac{r^{\prime} - R_{1}}{r^{\prime}}}}{ɛ_{\theta^{\prime}} = {\mu_{\theta^{\prime}} = \frac{r^{\prime}}{( {r^{\prime} - R_{1}} )}}}{ɛ_{Z^{\prime}} = {\mu_{Z^{\prime}} = {\lbrack \frac{R_{2}}{R_{2} - R_{1}} \rbrack^{2}\frac{r^{\prime} - R_{1}}{r^{\prime}}}}}} & ({A16})\end{matrix}$

where we have assumed that the starting material is vacuum.

FIGS. 4a and 4b show that the incident rays 404, 406 are diverted arounda central region 400, 402 and emerge from the cloaking region 408, 410apparently unperturbed, as predicted by the theoretical solution.

Whilst the theoretical solution has been confirmed by ray-tracingmethods, in order to put the invention into effect a suitable materialto form the cloaking structure has been developed.

The material parameters of the cloaking structure require that thecomponents of ∈′ equal those of μ′, that ∈′ and μ′ vary throughoutspace, and that the tensor components of ∈′ and μ′ vary independently.The anticipated material is thus inhomogeneous and anisotropic.Moreover, the material parameters ∈′ and μ′ must assume values less thanunity and approach zero at the interface between the concealed regionand the cloaking shell. Furthermore, it is noted that equation (6) issingular at r′=R₁. This is unavoidable as can be seen by considering aray headed directly towards the center of the sphere 100 (FIG. 1). Thisray does not know whether to be deviated up or down, left or right.Neighboring rays 104 are bent around tighter and tighter arcs closer tothe critical ray they are. This in turn implies very steep gradients in∈′ and μ′. Parameters ∈′ and μ′ are necessarily anisotropic in thecloaking material because space has been compressed anisotropically.This set of constraints is not attainable by conventional materials forexample because finding matching values for ∈ and μ in the same wavebanddoes not occur. However, given the rapid progress that has occurred overthe past several years in artificial materials, a metamaterial can bepractically designed that satisfies the specifications.

Metamaterials are artificially constructed ‘materials’ which can exhibitelectromagnetic characteristics that are difficult or impossible toachieve with conventional materials. From an electromagnetic point ofview, the wavelength, Δ, of a wave passing through a material determineswhether a collection of atoms or other objects can be considered amaterial and properties at the atomic level determine ∈ and μ. However,the electromagnetic parameters ∈ and μ need not arise strictly from theresponse of atoms or molecules: Any collection of objects whose size andspacing are much smaller than Δ can be described by an ∈ and μ. In thatcase, the values of ∈ and μ are determined by the scattering propertiesof the structured objects. Although such an inhomogeneous collection maynot satisfy an intuitive definition of a material, an electromagneticwave passing through the structure cannot tell the difference, and, fromthe electromagnetic point of view, we have created an artificialmaterial, or metamaterial. These properties of metamaterials areexplained further in “Metamaterials and Refractive Index”, D R Smith, JB Pendry, MCK. Wiltshire, VOL 305, SCIENCE 061081 (2004), which isincorporated herein by reference.

One useful property exhibited by certain metamaterials is that ofartificial magnetism. It has been found that structures consisting ofnon-magnetic arrays of wire loops in which an external field can inducea current consequently produce an effective magnetic response. One suchdesign is the so-called ‘Swiss-roll’ structure, as shown in FIG. 5. Theroll 500 is manufactured by rolling an insulated metallic sheet around acylinder 504. A design with about 11 turns on a 1-cm-diameter cylindergives a resonant response at 21 MHz. The metamaterial is formed bystacking together many of these cylinders. In this structure, the coiledcopper sheets have a self-capacitance and self-inductance that create aresonance. The currents that flow when this resonance is activatedcouple strongly to an applied magnetic field, yielding an effectivepermeability that can reach quite high values. No current can flowaround the coil except by self capacitance. As is shown in FIG. 6, whena magnetic field parallel to the cylinder 600 is switched on it inducescurrents (j) in the coiled sheets 602, which are spaced a distancedapart from one another. Capacitance between the first 604 and last 606turns of the coil enables current to flow. This is described further in“Magnetism from Conductors and Enhanced Non-Linear Phenomena”, J BPendry, A J Holden, D J Robbins and W J Stewart IEEE Trans. Micr. Theoryand Techniques, 47, 2075 (1999), which is incorporated herein byreference.

One problem with the Swiss-roll structures is that the continuouselectrical path provided by metal cylinders can cause the metamaterialto respond like an effective metal when the electric field applied isnot parallel to the cylinders, hence restricting its usefulness incertain applications. An adaptation described in the same paper whichavoids this undesirable effect is the split ring resonator (SRR). TheSRR is built up from a series of split rings, as shown in FIG. 7. Eachsplit ring 700 comprises at least two concentric thin metal rings 702,704 of width c, spaced apart by a distance d, each one with a gap, 706,708. By eliminating the continuous conducting path which the cylindersprovide, an SRR eliminates most of the electrical activity along thisdirection.

As shown in FIG. 8, it is possible to form a planar split ring structure800 from a series of split rings 802, spaced a distance a apart. The twodimensional square array 800 shown in FIG. 8 can be made by printingwith metallic inks. If each printed sheet is then fixed to a solid blockof inert material, thickness a, the blocks can be stacked to givecolumns of rings. This would establish magnetic activity along thedirection of stacking, z-axis.

The unit cell 900 of the stacked SRR structure is shown in FIG. 9 on theleft. It is possible to form a symmetrical three dimensional structurestarting from such a structure comprising successive layers of ringsstacked along the z-axis. This is achievable by cutting up the structureinto a series of slabs thickness a, making incisions in the y-z planeand being careful to avoid slicing through any of the rings. Each of thenew slabs contains a layer of rings but now each ring is perpendicularto the plane of the slab and is embedded within. The next stage is toprint onto the surface of each slab another layer of rings and stack theslabs back together again. The unit cell 902 of this second structure isshown in the middle of FIG. 9. In the final step, a third set of slabsis produced by cutting in the x-z plane, printing on the surface of theslabs, and reassembling. This results in a structure with cubic symmetrywhose unit cell 904 is shown on the right of FIG. 9.

Similarly c can be governed by an array of thin wires as described in“Extremely Low Frequency Plasmas in Metallic Mesostructures”, J BPendry, A J Holden, W J Stewart, I Youngs, Phys Rev Lett, 76, 4773(1996) which is incorporated herein by reference.

Hence known techniques can be applied to construct the metamaterialrequired to put the current invention as described above into practice.The materials comprise many unit cells, each of which is designed tohave a predetermined value for ∈′ and μ′, such that they correspond tothe values determined by equation (4) above (or for the non-orthogonalcase, equation G23 above. As will be apparent to the skilled reader,this may involve each unit cell having slightly different values of E′and to those of its radially neighboring cells. Each unit cell in themetamaterial acts as a concealing volume element. The concealing volumeelements are assembled in the correct respective spatial positions toform the cloaking structure, as calculated by the methods outlinedabove. The cloaking structure will, as a result, possess the spatiallydistributed material parameter values required for electromagneticconcealment of the enclosed volume.

For practical implementation of the theory outlined above, a threedimensional cloaking structure will be necessary in most cases. A solidcomposite metamaterial structure can be formed using fabricationtechniques similar to those used to manufacture a gradient index lens.One such known structure 1000 is shown in FIG. 10, in which SRR's andwires are deposited on opposite sides lithographically on a standardcircuit board to provide a desired material response. This is describedin more detail in “Reversing Light with Negative Refraction”, JB Pendry,DR Smith, Physics Today, June 2004, which is incorporated herein byreference.

The unit cell structure of the metamaterial used ensures that anisotropyand continuous variation of the parameters, as required by thetheoretical solution, can be achieved. Unrelated to cloaking, anotherapplication of the transformation theory described above is a device forconcentrating fields. FIG. 11 shows a concentrator that concentrateselectromagnetic fields incident upon its outer surface (110) onto aninner core region (114). The transformation that characterizes theconcentrator, for the spherical geometry, is shown in FIG. 11. The outerradius of the device (110) is R₃. A concentrating volume of space (112)with radius R₂<R₃ is compressed to a collecting volume (114) of radiusR₁. The shell volume (116) that lies between R₂ and R₃ is expanded tofill the region (118) between R₁ and R₃, as FIG. 11 depicts. A simpleexample of this transformation that compresses the core (112) uniformlyand stretches the shell region (116) with a linear radial function isgiven by

$\begin{matrix}{r = \{ \begin{matrix}{\frac{R_{2}}{R_{1}}r^{\prime}} & {0 < r^{\prime} < R_{1}} \\{{\frac{R_{2} - R_{1}}{R_{3} - R_{1}}R_{3}} + {\frac{R_{3} - R_{2}}{R_{3} - R_{1}}r^{\prime}}} & {R_{1} < r^{\prime} < R_{3}} \\r^{\prime} & {R_{3} < r^{\prime}}\end{matrix} } & (12) \\{{\theta = \theta^{\prime}}{\varphi = \varphi^{\prime}}} & (13)\end{matrix}$

The resulting material properties of permitting and permeability in theconcentration device are then

$\begin{matrix}{\lbrack {ɛ_{r^{\prime}}^{\prime},ɛ_{\theta^{\prime}}^{\prime},ɛ_{\varphi^{\prime}}^{\prime}} \rbrack = {\lbrack {\mu_{r^{\prime}}^{\prime},\mu_{\theta^{\prime}}^{\prime},\mu_{\varphi^{\prime}}^{\prime}} \rbrack = \{ \begin{matrix}{\frac{R_{2}}{R_{1}}\lbrack {1,1,1} \rbrack} & {0 < r^{\prime} < R_{1}} \\{\frac{R_{3} - R_{2}}{R_{3} - R_{1}}\lbrack {( {1 + {\frac{R_{2} - R_{1}}{R_{3} - R_{2}}\frac{R_{3}}{r^{\prime}}}} )^{2},1,1} \rbrack} & {R_{1} < r^{\prime} < R_{3}} \\\lbrack {1,1,1} \rbrack & {R_{3} < r^{\prime}}\end{matrix} }} & (15)\end{matrix}$

where the material properties are given relative to the externalenvironment. As with the cloaking structure, the external environmentmay be free space or any other medium.

In order to concentrate fields into an arbitrarily small volume, we letR₁→0 and obtain

$\begin{matrix}{\lbrack {ɛ_{r^{\prime}}^{\prime},ɛ_{\theta^{\prime}}^{\prime},ɛ_{\varphi^{\prime}}^{\prime}} \rbrack = {\lbrack {\mu_{r^{\prime}}^{\prime},\mu_{\theta^{\prime}}^{\prime},\mu_{\varphi^{\prime}}^{\prime}} \rbrack = \{ \begin{matrix}{\frac{R_{3} - R_{2}}{R_{3}}\lbrack {( {1 + {\frac{R_{3}}{R_{3} - R_{2}}\frac{R_{2}}{r^{\prime}}}} )^{3},1,1} \rbrack} & {0 < r^{\prime} < R_{3}} \\\lbrack {1,1,1} \rbrack & {R_{3} < r^{\prime}}\end{matrix} }} & (16)\end{matrix}$

If we wish the concentration cross section of the sphere to equal itsouter radius, then we let R₂→R₃.

$\begin{matrix}{{\lim_{R_{2}arrow R_{3}}{\lbrack {ɛ_{r^{\prime}}^{\prime},ɛ_{\theta^{\prime}}^{\prime},ɛ_{\varphi^{\prime}}^{\prime}} \rbrack {\lim_{R_{2}arrow R_{3}}\lbrack {\mu_{r^{\prime}}^{\prime},\mu_{\theta^{\prime}}^{\prime},\mu_{\varphi^{\prime}}^{\prime}} \rbrack}}} = \{ \begin{matrix}\lbrack {{+ \infty},0,0} \rbrack & {0 < r^{\prime} < R_{3}} \\\lbrack {1,1,1} \rbrack & {R_{3} < r^{\prime}}\end{matrix} } & (17)\end{matrix}$

This is the perfect field concentrator. The material parameters whichare required in the concentrator can be achieved through the use ofmetamaterials as described in relation to the cloaking structure. Thepermittivity ∈ and permeability μ can be controlled independently of oneanother in each of three dimensions within each unit cell of themetamaterial.

The concentrator device is unique, and different from focusing lensesand mirrors, in that it operates on fields incident from any direction.For electromagnetic sensing or energy harvesting applications, adetector or collector would be placed at the core (114) to interact withthe intensified fields at that location. Part of the utility of thisdevice is concentration of fields when the direction of their source isinitially unknown or constantly changing e.g. solar energy collection ormicrowave energy beaming on mobile platforms.

A detector placed at the concentrator's core (114) could easily beconfigured to provide accurate directional information as well ascollect the energy. This could be accomplished by, for example, using aspherical shaped detector patterned into eight quadrants. By comparingthe energy collected from each quadrant, accurate directionalinformation will be obtained. The detector need not be spherical. Atransformation could be used that results in a core region that is athin circular disk. This would make the concentrator compatible withcommon planar detectors.

The device can also provide impedance matching to the detector. This isachieved by applying the inverse transformation to the desired detector,expanding it from the core radius, R₁, to the radius R₂. It is thenpossible to design an impedance matching layer, (between R₁ and R₃), tomatch the transformed detector to its environment. This impedancematching layer is then transformed with the forward transformationexpanding it to fill the region (118) between R₁ and R₃. The resultingshell layer between R₁ and R₃ provides both concentration and impedancematching functions.

Like the cloak structures the concentrator (110) can be perfectlymatched to its environment, so that if the fields are not disturbed inthe core (114) by a detector, the device (110) is undetectable byexternal electromagnetic probing. This device (110) can operate in bothnear and far field regimes and at any size scale and wavelength. Thebasic formalism of this transformation can be applied to other basicgeometries, (e.g. cylindrical) or applied conformally to an arbitrarilyshaped object. FIG. 12 demonstrates the concentration of rays or fieldlines from a point source (120) incident upon a sphere (122) radius R₃into an inner core region (124) radius R₁.

The co-ordinate transfer mapping theory outlined above has beendescribed with respect to diverting the propagation of electromagneticwaves. However it is noted that there is a significant analogy betweenelectromagnetic and other propagating waves such as acoustic waves. Inparticular, this theory may be applied to obtain specifications for themechanical properties of a material such as density and stiffness thatyield acoustic cloaking. Such a cloak may be used for example to concealbodies from sonar detection.

Whilst the description of the cloaking theory above is directed mainlyto concealing objects, it will be appreciated that the diversion ofelectromagnetic disturbances from the cloaked volume into a cloakingshell creates an electromagnetic quiet zone inside. Equation (9)dictates that any electromagnetic radiation produced by an object insidethe quiet zone will not be able to emerge from it. This may be put touse in electromagnetic interference shielding applications, for exampleinside Mill machines. Analogously, acoustic quiet zones may also beachieved.

The application of the present invention is not limited to the cloakingand concentrator structures as shown in the Figures. As demonstratedclearly in above, the mathematical solution provided herein applies tocloak a structure of any shape. Furthermore, if a volume is cloaked, anobject of any shape or material placed therein will not be detectable byan outside observer, either in reflection mode or transmission mode.

The methods described herein can be used to apply any appropriatecoordinate transformation between any appropriate co-ordinate systemsand using any appropriate function of the co-ordinates in the originalframe. The cloaking structure used to conceal the material within it canbe of any shape or thickness. In particular, the cloaking structure doesnot have to follow the same shape as the object within that is to beconcealed. It will be apparent to the skilled reader that the methodsdescribed herein can be used to transform all frequencies ofelectromagnetic radiation, and for a given application can be applied toradiation in any specific frequency range of the electromagneticspectrum, including infrared, radio, visible light, x-rays, microwavesand radar.

In the examples shown impedance matching has been achieved by the ratio

$\frac{\mu}{ɛ} = 1$

As will be apparent to the skilled reader, impedance matching of thesort described herein can be achieved for

$\frac{\mu}{ɛ} = m$

where m is any real constant value. In particular, the desired ratio ofμ/∈ will depend on the medium in which the cloaking structure is to beembedded.

The reference J. B. Pendry et al., “Magnetism from Conductors andEnhanced Nonlinear Phenomena,” IEEE Trans. Micro. Theory Tech. 47(1999), 2074-2084, above incorporated by reference, includes thefollowing text:

C “Swiss Roll” Capacitor

In this instance, we find for the effective permeability

$\begin{matrix}{\mu_{eff} = {{1 - \frac{F}{1 + \frac{2\sigma \; i}{\omega \; r\; {\mu_{0}( {N - 1} )}} - \frac{1}{2\pi^{2}r^{3}{\mu_{0}( {N - 1} )}^{2}\omega^{2}c}}} = {1 - \frac{\frac{\pi \; r^{2}}{a^{2}}}{1 + \frac{2\sigma \; i}{\omega \; r\; {\mu_{0}( {N - 1} )}} - \frac{{dc}_{0}^{2}}{2\pi^{2}r^{3}{\mu_{0}( {N - 1} )}\omega^{2}}}}}} & (29)\end{matrix}$

where F is as before the fraction of the structure not internal to acylinder, and the capacitance per unit area between the first and thelast of the coils is

$\begin{matrix}{C = {\frac{ɛ_{0}}{d( {N - 1} )} = \frac{1}{\mu_{0}{{dc}_{0}^{2}( {N - 1} )}}}} & (30)\end{matrix}$

IV. An Isotropic Magnetic Material

. . . . We propose an adaptation of the “split ring” structure, in whichthe cylinder is replaced by a series of flat disks each of which retainsthe “split ring” configuration, but in slightly modified form . . . .

The effective magnetic permeability we calculate, on the assumption thatthe rings are sufficiently close together and that the magnetic lines offorce are due to currents in the stacked rings, are essentially the sameas those in a continuous cylinder. This can only be true if the radiusof the rings is of the same order as the unit cell side. We arrive at

$\mu_{eff} = {{1 - \frac{\frac{\pi \; r^{2}}{a^{2}}}{1 + {\frac{2l\; \sigma_{1}}{\omega \; r\; \mu_{0}}i} - \frac{3\; l}{\pi^{2}\mu_{0}\omega^{2}C_{1}r^{3}}}} = {1 - \frac{\frac{\pi \; r^{2}}{a^{2}}}{1 + {\frac{2l\; \sigma_{1}}{\omega \; r\; \mu_{0}}i} - \frac{3\; {lc}_{0}^{2}}{{\pi\omega}^{2}\ln \frac{2\; c}{d}r^{3}}}}}$

where σ₁ is the resistance of unit length of the sheets measured aroundthe circumference.

The reference J. B. Pendry et al., “Extremely Low Frequency Plasmons inMetallic Mesostructures,” Phys. Rev. Lett 76 (1996), 4773-4776, aboveincorporated by reference, includes the following text:

The plasmons have a profound impact on properties of metals, not leaston their interaction with electromagnetic radiation where the plasmonproduces a dielectric function of the form

$\begin{matrix}{{ɛ(\omega)} = {1 - \frac{\omega_{p}^{2}}{\omega ( {\omega + {i\; \gamma}} )}}} & (2)\end{matrix}$

which is approximately independent of wave vector, and the parameter γis a damping term representing dissipation of the plasmon's energy intothe system . . . .

In this Letter we show how to manufacture an artificial material inwhich the effective plasma frequency is depressed by up to 6 orders ofmagnitude. The building blocks of our new material are very thinmetallic wires . . . .

Having both the effective density, n_(eff), and the effective mass,m_(eff), on hand we can substitute into (1),

$\begin{matrix}{\omega_{p}^{2} = {\frac{n_{eff}e^{2}}{ɛ_{0}m_{eff}} = {\frac{2\pi \; c_{0}^{2}}{a^{2}{\ln ( \frac{a}{r} )}} \approx ( {8.2\mspace{14mu} {GHz}} )^{2}}}} & (14)\end{matrix}$

Here is the reduction in the plasma frequency promised.

Note in passing that, although the new reduced plasma frequency can beexpressed in terms of electron effective mass and charge, thesemicroscopy quantities cancel, leaving a formula containing onlymacroscopic parameters of the system: wire radius and lattice spacing.

While this invention has been particularly shown and described withreferences to preferred embodiments thereof, it will be understood bythose skilled in the art that various changes in form and details may bemade therein without departing from the scope of the inventionencompassed by the appended claims

What is claimed is:
 1. A method of constructing a diverting volume fordiverting propagation of an incident electromagnetic wave in a specificfrequency range below visible light frequencies around a core volume,the incident electromagnetic wave having an original propagationdirection, said method comprising: identifying a coordinate transformthat diverts the propagation direction from within the core volume tothe diverting volume; using the identified coordinate transform,computing material parameters of the diverting volume; selectingdiverting-volume elements to match the computed material parameters ofthe diverting volume for the specific frequency range; and constructingthe diverting volume by arranging the selected diverting-volume elementsaround the core volume, wherein the selected diverting-volume elementsare arranged so as to return the incident electromagnetic wave to itsoriginal propagation direction.